1.1 Background of Study
Consumer Price Index (CPI) is a measure of the average change overtime in the prices of goods and services that people buy for day-to-day living. The CPI uses data from survey of consumption pattern of households to produce a timely and precise average price change for the consumption sector of any economy like the Nigerian economy. The most important task in the production of CPI is the determination of the market basket of goods and services whose current price is usually compared with its base year price to measure changes in price.

Annual CPI has pattern in the forms of small local fluctuations and cyclic movements that have been recognized over the years. Upward or downward movements in CPI sometimes persist for a very long period and also tend to retrace back and forth as the case may be. As a result, world economies witness periods of boom and recession or downturn. In order to understand what causes this, local fluctuations and cyclic movements can be modeled in a variety of ways in accordance with Mooney et al (2006). In this study we describe and compare two of these possibilities. Notwithstanding that this work was on Nigeria Consumer Price Index (NCPI) data, the comparisons are also relevant to other situations where data contain a seasonal, or other cyclic, pattern, for example in Geology (Upton et al., 2003), biology (Batschelet, 1981) and atmospheric science (Wilks, 1995, Section 8.4).

The approach to modeling and forecasting time series such as the Nigeria Consumer Price Index(NCPI) data has been developed over several decades (see Grenander and Rosenblatt, 1957; Box and Jenkins, 1976; Makridakis and Wheelwright, 1993). The modeling and forecasting methods for both stationary and non-stationary time series have been applied to many different fields and many successful results have been obtained in different areas. The method developed by Box, Jenkins and Reinsel (2008,p.78); that based on the modeling of time series by autoregressive integrated moving average (ARIMA) processes, is one of the two approaches in this work.
The second approach is the frequency domain approach, which assumes that the time series is best regarded as a sum or linear superposition of periodic sine and cosine waves of different periods or frequencies. See, for example, Pollock (1999, p. 555) and Khuri (2003, p. 500).

Statement of the Problem
In this work, we shall fit different ARIMA models to the Nigeria Consumer Price Index (NCPI) data. We shall then compare the ARIMA models. The error levels of the models will be critically assessed to select the best which will guide prediction and policy formulation and implementation in Nigeria.

1.3 Objective of the Study
We shall build time series model for time and frequency domain that is Box-Jenkins ARIMA model. Specifically, the objectives of the study are:
To fitdifferent Box-Jenkins ARIMA models to the Nigeria Consumer Price Index (NCPI) data.
To test using AKAIKE’S Information Criterion to check the best fitting model.
To forecast the inflation rate for some lead time using the best model.

1.4. Significance of the Study
The significance of this study would help in knowing if there is any increasing or decreasing change in the trend of Nigeria Consumer Price Index (NCPI).
This will also help in planning for the uncertain future and how to allocate resources for Nigerian Consumers. It will help producers to know when to produce more or less, to meet consumers’ demand.
Lastly, this research will serve as a reference material and guide to others who may wish to embark on the same field of study.
1.5 Scope and Limitation of the Study
This research is limited to the National Bureau of Statistics of Nigeria; it studies the trend of Nigeria All Items Less Farm Produce Consumer Price Index and the data is based on the outcome from January, 2009 to May, 2017.

1.6 Definition of Terms
AR (Autoregressive): A model in which future values are forecast purely on the basis of past values of the time series.
MA (Moving Average): A model in which future values are forecast purely on the basis of past shocks (or noise or random disturbances).
ARMA (Autoregressive Moving Average): A model that uses both past values of the time series and past shocks.
ARIMA (Autoregressive Integrated Moving Average): An ARMA model of a differenced series is called an ARIMA model, where the ‘I’ stand’s for Integrated because the output needs to be anti-differenced or integrated, to forecast the original series.
Stationarity: This means that the properties of the series do not depend on the time when it is captured.
Autocorrelation Function (ACF): Measures the ratio of the covariance between observations k lags apart and the geometric average of the variance of observations (i.e., the variance of the process is stationary as V(Xt) = V(Xt-k). The autocorrelation of a stationary time series is the correlation between current and past values. Given a series Xt, …, Xn the autocorrelation at lag k, k = 0,1,2,3,…. is estimated thus
………………………………………………………… (1)
Where =
The autocorrelation coefficient is given by
Partial Autocorrelation Function (PACF):
The partial autocorrelation function measures the correlation between a particular lag and the current value after the effect of the other lags has been predicted. The partial auto correlation function (PACF) is given as a coefficient of a sample autoregressive process of the form according to Durbin 1960
…………………………………………………………… (2)
As explained by Box and Jenkins, the sample ACF plot is useful in determining the type of model to fit to a time series of length N. Since ACF is symmetrical about lag zero, it is only required to plot the sample ACF for positive lags, from lag one onwards to a maximum lag of about N/4. The sample PACF plot helps in identifying the maximum order of an AR process.
Forecasting: This is the use of model to predict future values based on previously observed values.
Time Series Analysis: Time Series Analysis comprises of the method for analyzing time series data in order to extract meaningful statistics and other characteristics of data.
White Noise: The white noise process is a sequence of uncorrelated random variables with mean zero and constant variance, E( = 0 and Var( =
A process et may be regarded as a series of shocks which drives the system, that is
………………………………………………………………… (3)